Finite Group Databasebeta

← #10702876

Subgroups — #10702876

Order 768.

1110 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#99C211yesmaximal, characteristic
2#84043C211yesmaximal, characteristic
3C22×#834C211yesmaximal, characteristic
4C2×#4527C212yesmaximal, complemented
5C2×#4526C212yesmaximal, complemented
6#11169—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C22×C3×C22⋊C4C2212yes—
9C22×#217C2211yescharacteristic
10C22×#217C2211yescharacteristic
11C22×#217C2212yes—
12C2×C3×#99—22no—
13C2×C3×#99—24no—
14#5247C412yescomplemented
15#5220—22no—
16#5220—22no—
17C2×#834—22no—
18C2×#834—24no—
19#5211C412yescomplemented
20#4527—24no—
21#4526—24no—
22C22×#99—31no—
23C22×#99D311yescharacteristic
24#2308—31no—
25C2×#342—32no—
26C2×#342—32no—
27C25×C3C2×C411yescharacteristic
28C25×C3C2311yescharacteristic
29C22×C3×D4—22no—
30C22×C3×D4—22no—
31C22×C3×D4—22no—
32C22×C3×D4—22no—
33C22×C3×D4—22no—
34C22×C3×D4—22no—
35C22×C3×D4D412yes—
36C22×C3×D4D412yes—
37C23×C3×C4—22no—
38C23×C3×C4C2×C411yescharacteristic
39C23×Dic3—21no—
40C23×Dic3—21no—
41C23×Dic3—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4D414yes—
48C2×#217—22no—
49C2×#217—22no—
50C2×#217—22no—
51C2×#217—22no—
52C2×#217—22no—
53C2×#217—22no—
54C2×#217—22no—
55C2×#217—22no—
56C2×#217—21no—
57C2×#217—21no—
58C2×#217—22no—
59C2×#217—24no—
60C2×#217C2×C412yes—
61C2×#217C2×C412yes—
62C2×#217D414yes—
63C3×#99—28no—
64#834—28no—
65C3⋊#95—24no—
66C23×D4C2×D311yescharacteristic
67C22×C22⋊C4—31no—
68C22×C22⋊C4—31no—
69C22×C22⋊C4—32no—
70C22×C22⋊C4C2×D312yes—
71C2×#99—22no—
72C2×#99—24no—
73C2×#99—64no—
74C2×#99—62no—
75#386—62no—
76#386—62no—
77#381—32no—
78#379—32no—
79#342—64no—
80#342—64no—
81C24×C3—21no—
82C24×C3—21no—
83C24×C3—22no—
84C24×C3—22no—
85C24×C3—82no—
86C24×C3—82no—
87C24×C3C22⋊C412yes—
88C24×C3C22×C411yescharacteristic
89C24×C3C2×D411yescharacteristic
90C24×C3C2×D412yes—
91C24×C3C2×D411yescharacteristic
92C24×C3C2×D412yes—
93C24×C3#4112yes—
94C24×C3#4112yes—
95C24×C3#4112yes—
96C24×C3#4112yes—
97C2×C3×D4—22no—
98C2×C3×D4—22no—
99C2×C3×D4—24no—
100C2×C3×D4—24no—
101C2×C3×D4—24no—
102C2×C3×D4—22no—
103C2×C3×D4—22no—
104C2×C3×D4—24no—
105C2×C3×D4—42no—
106C2×C3×D4—42no—
107C2×C3×D4—42no—
108C2×C3×D4—42no—
109C2×C3×D4—42no—
110C2×C3×D4—42no—
111C2×C3×D4—42no—
112C2×C3×D4—42no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—24no—
118C22×C3×C4—42no—
119C22×C3×C4—42no—
120C22×C3×C4—84no—
121C22×C3×C4C22⋊C412yes—
122C22×Dic3—22no—
123C22×Dic3—22no—
124C22×Dic3—24no—
125C22×Dic3—42no—
126C22×Dic3—42no—
127C22×Dic3—42no—
128C22×Dic3—42no—
129C22×Dic3—82no—
130C22×Dic3—82no—
131C22×Dic3—84no—
132C3×C22⋊C4—24no—
133C3×C22⋊C4—24no—
134C3×C22⋊C4—42no—
135C3×C22⋊C4—42no—
136#217—24no—
137#217—24no—
138#217—24no—
139#217—24no—
140#217—41no—
141#217—41no—
142#217—42no—
143#217—42no—
144#217—41no—
145#217—41no—
146#217—84no—
147C25C4×D311yescharacteristic
148C25C22×D311yescharacteristic
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4D1212yes—
156C22×D4#6712yes—
157C23×C4—22no—
158C23×C4—61no—
159C23×C4—62no—
160C23×C4—61no—
161C23×C4C4×D311yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—32no—
168C2×C22⋊C4—32no—
169C2×C22⋊C4—34no—
170C2×C22⋊C4—62no—
171C2×C22⋊C4—62no—
172C2×C22⋊C4—62no—
173C2×C22⋊C4—62no—
174C2×C22⋊C4—62no—
175C2×C22⋊C4—62no—
176C2×C22⋊C4—62no—
177C2×C22⋊C4—62no—
178C2×C22⋊C4—61no—
179C2×C22⋊C4—62no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—64no—
182C2×C22⋊C4#6714yes—
183#99—28no—
184#99—68no—
185#95—64no—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3—22no—
189C23×C3—42no—
190C23×C3—42no—
191C23×C3—44no—
192C23×C3—44no—
193C23×C3—42no—
194C23×C3—44no—
195C23×C3—42no—
196C23×C3—42no—
197C23×C3—82no—
198C23×C3—82no—
199C23×C3—82no—
200C23×C3—82no—
201C23×C3—82no—
202C23×C3—82no—
203C23×C3—84no—
204C23×C3—84no—
205C23×C3C2×C22⋊C411yescharacteristic
206C23×C3#11711yescharacteristic
207C23×C3#11711yescharacteristic
208C23×C3C4×D412yes—
209C23×C3C4×D412yes—
210C23×C3#12011yescharacteristic
211C23×C3#12011yescharacteristic
212C23×C3#12112yes—
213C23×C3#12112yes—
214C23×C3#12112yes—
215C23×C3#12112yes—
216C23×C3#12312yes—
217C23×C3#12312yes—
218C23×C3#12311yescharacteristic
219C23×C3#12311yescharacteristic
220C23×C3#12312yes—
221C23×C3#12312yes—
222C23×C3#12412yes—
223C23×C3#12412yes—
224C23×C3#12612yes—
225C23×C3#12612yes—
226C3×D4—44no—
227C3×D4—44no—
228C3×D4—44no—
229C3×D4—44no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—44no—
236C2×C3×C4—84no—
237C2×C3×C4—84no—
238C2×C3×C4—84no—
239C2×Dic3—44no—
240C2×Dic3—44no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—44no—
244C2×Dic3—84no—
245C2×Dic3—84no—
246C2×Dic3—84no—
247C2×Dic3—84no—
248C2×Dic3—84no—
249C24—21no—
250C24—21no—
251C24—22no—
252C24—22no—
253C24—82no—
254C24—82no—
255C24(C2×D3)⋊C412yes—
256C24C2×C4×D311yescharacteristic
257C24C2×D1211yescharacteristic
258C24#23512yes—
259C24D3×D412yes—
260C24#23712yes—
261C24#23712yes—
262C24#23712yes—
263C24C2×#6711yescharacteristic
264C24C2×#6712yes—
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—62no—
289C22×C4—62no—
290C22×C4—64no—
291C22×C4—84no—
292C22×C4—122no—
293C22×C4—122no—
294C22×C4—122no—
295C22×C4—122no—
296C22×C4—242no—
297C22×C4—242no—
298C22×C4—244no—
299C22×C4(C2×D3)⋊C412yes—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—64no—
305C22⋊C4—64no—
306C22⋊C4—64no—
307C22⋊C4—64no—
308C22⋊C4—121no—
309C22⋊C4—121no—
310C22⋊C4—121no—
311C22⋊C4—122no—
312C22⋊C4—122no—
313C22⋊C4—121no—
314C22⋊C4—244no—
315C22×C3—22no—
316C22×C3—24no—
317C22×C3—24no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—42no—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—42no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—42no—
334C22×C3—42no—
335C22×C3—42no—
336C22×C3—44no—
337C22×C3—42no—
338C22×C3—82no—
339C22×C3—82no—
340C22×C3—84no—
341C22×C3—84no—
342C22×C3#38111yescharacteristic
343C22×C3#38611yescharacteristic
344C22×C3#38611yescharacteristic
345C22×C3#38812yes—
346C22×C3#38812yes—
347C22×C3#39412yes—
348C22×C3#39412yes—
349C22×C3#39712yes—
350C22×C3#39712yes—
351C3×C4—42no—
352C3×C4—84no—
353Dic3—82no—
354Dic3—82no—
355Dic3—84no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—44no—
360C23—42no—
361C23—44no—
362C23—42no—
363C23—42no—
364C23—44no—
365C23—42no—
366C23—42no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23—84no—
374C23—84no—
375C23#87711yescharacteristic
376C23#87711yescharacteristic
377C23#87912yes—
378C23#87912yes—
379C23#67⋊C412yes—
380C23#88211yescharacteristic
381C23#88312yes—
382C23#88512yes—
383C23#88612yes—
384C23#88611yescharacteristic
385C23C2×(C2×D3)⋊C411yescharacteristic
386C23C4×#6712yes—
387C23#92911yescharacteristic
388C23#93012yes—
389C23#93512yes—
390C23#93512yes—
391C23#93612yes—
392C23#93812yes—
393C23#93912yes—
394C23#93912yes—
395C23#95311yescharacteristic
396D4—44no—
397D4—44no—
398D4—44no—
399D4—44no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—124no—
410C2×C4—124no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—124no—
414C2×C4—244no—
415C2×C4—244no—
416C2×C4—244no—
417C2×C4—244no—
418C2×C4—244no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—22no—
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—22no—
425C2×C3—22no—
426C2×C3—22no—
427C2×C3—82no—
428C2×C3#144011yescharacteristic
429C2×C3#182812yes—
430C22—24no—
431C22—22no—
432C22—24no—
433C22—22no—
434C22—22no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—41no—
446C22—42no—
447C22—41no—
448C22—41no—
449C22—42no—
450C22—41no—
451C22—42no—
452C22—44no—
453C22—82no—
454C22—82no—
455C22—84no—
456C22—84no—
457C22#494311yescharacteristic
458C22#494412yes—
459C22#494512yes—
460C22#495212yes—
461C22#495312yes—
462C22#495811yescharacteristic
463C22#496112yes—
464C22#521311yescharacteristic
465C22#522212yes—
466C4—42no—
467C4—84no—
468C4—242no—
469C4—242no—
470C4—244no—
471C3#1116911yescharacteristic, complemented, Sylow
472C2—22no—
473C2—21no—
474C2—21no—
475C2—22no—
476C2—22no—
477C2—22no—
478C2—22no—
479C2—22no—
480C2—82no—
481C2#7002012yes—
482C2#7154911yescharacteristic
subgroupG/H
centerC22#4943
commutatorC24×C3C22×C4
FrattiniC25C22×D3
FittingC22×C3×#99C2
socleC22×C3#381
prime radicalC23×C3×D4C22
cosocleC25×C3C23
hypercenterC22×#99D3
hypocenterC3#11169
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ C22×#99

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C2×#4527C2semidirect
5. C2×#4526C2semidirect
14. #5247C4semidirect
19. #5211C4semidirect
471. C3#11169semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3×#992
#840432
C22×#8342
C2×#45272
C2×#45262
#111693